What’s This Algorithm Stuff About Then ##TOP##

The use of algorithms is spreading as massive amounts of data are being created, captured and analyzed by businesses and governments. Some are calling this the Age of Algorithms and predicting that the future of algorithms is tied to machine learning and deep learning that will get better and better at an ever-faster pace.

A number of respondents noted the many ways in which algorithms will help make sense of massive amounts of data, noting that this will spark breakthroughs in science, new conveniences and human capacities in everyday life, and an ever-better capacity to link people to the information that will help them. They perform seemingly miraculous tasks humans cannot and they will continue to greatly augment human intelligence and assist in accomplishing great things. A representative proponent of this view is Stephen Downes, a researcher at the National Research Council of Canada, who listed the following as positive changes:

Whats This Algorithm Stuff AboutThen

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Participants in this study were in substantial agreement that the abundant positives of accelerating code-dependency will continue to drive the spread of algorithms; however, as with all great technological revolutions, this trend has a dark side. Most respondents pointed out concerns, chief among them the final five overarching themes of this report; all have subthemes.

The respondents to this canvassing offered a variety of ideas about how individuals and the broader culture might respond to the algorithm-ization of life. They argued for public education to instill literacy about how algorithms function in the general public. They also noted that those who create and evolve algorithms are not held accountable to society and argued there should be some method by which they are. Representative comments:

Write a function named collatz() that has one parameter named number. If the number is even, then collatz() should print number // 2 and return this value. If the number is odd, then collatz() should print and return 3 * number + 1. Then write a program that lets the user type in an integer and that keeps calling collatz() on that number until the function returns the value 1.

This is what I came up with for this practice exercise. It asks for an input Validates whether it's an integer. If not it rebukes and exits. If it is, it loops through the collatz sequence until the result is 1 and then you win.

Essentially, the Facebook algorithm evaluates every post, ad, Story, and Reel. It scores content and then arranges it in descending, non-chronological order of interest for each individual user. This process happens every time a user refreshes their feed.

The Facebook algorithm is a set of ranking signals powered by machine learning and artificial intelligence. It calculates which content is most likely to appeal to each user and then delivers them a personalized feed.

Let's imagine that you have an array of n items. If you perform a lookup in this array, then that lookup can return one of n + 1 values: either "the item isn't present," or "the item is present at index i" for any of the n indices.

Now, suppose that the only way that your algorithm is allowed to work with the array is by asking questions of the form "is the item greater than or equal to the item in index i?" for some choice of i, and let's imagine that you ask some question of this form k total times. There are then 2k possible ways that the comparisons could come back. To see why, there are two options for the how the first comparison can go (either "yes" or "no"). There are two options for how the second comparison can go (either "yes" or "no"), and two options for the third comparison. Multiplying all those 2s together gives 2k.

On the more theoretical side of things, fusion trees can perform searches in time O(log n / log w), where w is the machine word size, provided that the values are integers that fit into a single machine word. This can be improved down to the surprising runtime of O(sqrt(log n / log log n)). It's known that if the n values each fit into a single machine word, then the predecessor lower bound says you can't do better than the (very unusual runtime of) O(min{log w / log log w, sqrt(log n / log log n)}), where w is the machine word size. These algorithms outperform the (log n) lower bound by making multiple comparisons in parallel using creative operations on individual machine words.

To start with be careful about using the word "faster" when talking about Big-O complexity as it's done in the question title. Big-O says nothing about how fast an algorithm is. Big-O only tells how execution time changes when some variable N changes. Example:

Also notice that for some N values a O(N^2) algorithm may be faster than an O(N) algorithm. Big-O doesn't tell anything about that. All we can say is that if we keep increasing N then sooner or later the O(N) algorithm will be faster than the O(N^2) algorithm. But Big-O doesn't tell what that value of N is. It can be for N=1, N=10, N=100, ... So be careful about "translating" Big-O complexity into "fast".

The above covers the general array but notice that for some arrays the assumption may not hold. Sometimes an algorithm may extract more information from the look-up'ed value at index i. For instance if we know something about the distribution of values in the array and our search value is far from the look-up value, an algorithm may benefit from doing something else in the next look-up than doing the next look-up at N/2 and thereby be more efficient.

A remark: if you use two symbols (i.e. write integers in binary), and N = 2k (to simplify), then the dichotomic search algorithm that you surely know (or will know very soon) consists exactly in determining the k symbols (bits) of the output one by one, from the most significant bit to the least significant bit.

The only info we have is it's sorted. Let's say we are in the middle of the array now you make a decision based on 1 comparison that if the left half has the answer or the right half has the answer or mid itself is the answer and the recursively doing the comparison till we find the element ...The point to note here is that you are able to do so is because the array is sorted now let assume we are standing at some random point now still you can apply the same algorithm but you might partition the array unevenly which might screw up your number of comparison. Let say your random point is always at starting or ending point of the array the algo will have the complexity of O(N) in the worst-case. So, The best way to search for an element in the next recursion call is to reduce the search space by half each time which you are able to do in only 1 comparison, now you cannot decide this in less than 1 comparison unless there is some kind of pre-processing done which will again take more than O(log(N)

An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large. For example, an algorithm can be an algebraic equation such as y = m + n (i.e., two arbitrary "input variables" m and n that produce an output y), but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):

Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example".[57] But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computer must know how to take a square root. If they do not, then the algorithm, to be effective, must provide a set of rules for extracting a square root.[58]

But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, the arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".[59] When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement").

One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:

Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest.[68] While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm.

But "exceptional cases"[70] must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if negative numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates it) is a partial function rather than a total function. A notable failure due to exceptions is the Ariane 5 Flight 501 rocket failure (June 4, 1996). be457b7860